Mathematics 9709 · AS & A Level · Continuous random variables
Continuous random variables — practice question
Alethia represents the delay time, in minutes, of her train on any day by the random variable $X$, with probability density function
$f(x) = \begin{cases} \frac{3}{8000}(x - 20)^2, & 0 \le x \le 20, \\ 0, & \text{otherwise}. \end{cases}$
(a)[4]
Find the probability that the train is more than $10$ minutes late on both of two randomly selected days.
(b)[4]
Find the value of $\mathrm{E}(X)$.
(c)[4]
Let the median of $X$ be $m$. Show that $m$ satisfies the equation $(m - 20)^3 = -4000$, and then find $m$ correct to $3$ significant figures.
(d)[1]
State one respect in which Alethia’s model may be unrealistic.
Worked solution & mark scheme
This 13-mark question has a full step-by-step worked solution and mark scheme. One marking point: “Set up the expression $P(X > 10) = \int_{10}^{20} \dfrac{3}{8000}(x-20)^2\,dx$” …